Gorenstein–Harada theorem explained

In mathematical finite group theory, the Gorenstein–Harada theorem, proved by Daniel Gorenstein and Koichiro Harada, classifies the simple finite groups of sectional 2-rank at most 4.^{[1] [2]} It is part of the classification of finite simple groups.^[3]

Finite simple groups of section 2 with rank at least 5 have Sylow 2-subgroups with a self-centralizing normal subgroup of rank at least 3, which implies that they have to be of either component type or of characteristic 2 type. Therefore, the Gorenstein–Harada theorem splits the problem of classifying finite simple groups into these two sub-cases.

Notes and References

1. Book: Gorenstein . D. . Daniel Gorenstein . Harada . Koichiro . Koichiro Harada . Gagen . Terrence . Hale . Mark P. Jr. . Shult . Ernest E. . Finite groups '72. Proceedings of the Gainesville Conference on Finite Groups, March 23-24, 1972 . North-Holland . Amsterdam . North-Holland Math. Studies . 978-0-444-10451-9 . 0352243 . 1973 . 7 . Finite groups of sectional 2-rank at most 4 . 57–67.
2. Book: Gorenstein . D. . Harada . Koichiro . Finite groups whose 2-subgroups are generated by at most 4 elements . . Providence, R.I. . Memoirs of the American Mathematical Society . 978-0-8218-1847-3 . 0367048 . 1974 . 147.
3. Book: Bob Oliver. Reduced Fusion Systems over 2-Groups of Sectional Rank at Most 4 . 25 January 2016. . 978-1-4704-1548-8 . 1, 3.